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As I understand it, the definition of the hazard ratio is the ratio of two hazard rates. Often the exp(coef) from a Cox model is also used as an estimate of the hazard ratio. These methods give two different, although similar, results. Are there circumstances in which it's appropriate to use one method vs. the other?

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@ocram answer goes direct to the point (+1). However, you're talking about two different methods to estimate HRs, but then you mention only the Cox model. I'm curious to know what other method you use to estimate an HR. Because maybe what you're estimating using the method other than Cox model is not really an HR. –  andrea Apr 13 '12 at 11:09

2 Answers 2

I would suggest that you write the model you are working with, and then compute the hazard ratio you are interested in, in order to know how to estimate it from your $\hat{\beta}$'s.

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Example

Suppose your Cox's proportional hazards model takes the form $$h(t) = h_0(t) \exp(x_1 \beta_1 + x_2 \beta_2)$$ where $h_0$ is the baseline hazard function, and $\beta_1$ and $\beta_2$ are regression coefficients associated with the covariates $x_1$ and $x_2$. Suppose further that $x_1$ is binary ($0$ or $1$).

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$$\frac{h(t | x_1=1, x_2)}{h(t | x_1=0, x_2)} = \exp(\beta_1).$$ So, $\exp(\beta_1)$ is a conditional (given the value of $x_2$) hazard ratio ($x_1 = 1$ versus $x_1 = 0$).

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Thanks for the answer, I wasn't clear. I'm not comparing the Cox model's hazard ratio to another model's, but to the definition of the hazard ratio as simply the ratio of two hazard rates (-log(frac. surv pop1)*time)/(-log(frac.surv pop2)*time). I have seen both methods used when either could be applied, that is, where the cox model is univariate, so one variable isn't being assessed relative to another. –  Huazhong Apr 13 '12 at 12:12

In your comment you make clear that you are comparing the weighted instantaneous hazard ratio ( i.e., exp(beta)) with some summary measure of the ratios of cumulative hazards ( -log(frac.surv pop2)*time ). I think you will find that they are roughly the same during the early early phases of follow-up but they diverge as time increases. The reason for this is that as survival approaches zero the ratio of cumulative survival necessarily goes to unity, where as the hazard ratio may not.

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